You will probably need to use your file manager (Windows File Explorer or Mac Finder) to locate the playlist file (.m3u) and delete it. Generally the video, music and photo option in the Twonky Server My Library section is only for playing media not for managing (deleting) it.

On a side note, there are generally much better ways to play the media that Twonky has scanned than using Twonky’s Server web page. For example, one can use a DLNA client to play the Twonky scanned media.